The Stacks project

Lemma 15.62.13. Let $A \to B$ be a ring map. Let $K^\bullet $ be an $m$-pseudo-coherent (resp. pseudo-coherent) complex of $A$-modules. Then $K^\bullet \otimes _ A^{\mathbf{L}} B$ is an $m$-pseudo-coherent (resp. pseudo-coherent) complex of $B$-modules.

Proof. First we note that the statement of the lemma makes sense as $K^\bullet $ is bounded above and hence $K^\bullet \otimes _ A^{\mathbf{L}} B$ is defined by Equation (15.56.0.2). Having said this, choose a bounded complex $E^\bullet $ of finite free $A$-modules and $\alpha : E^\bullet \to K^\bullet $ with $H^ i(\alpha )$ an isomorphism for $i > m$ and surjective for $i = m$. Then the cone $C(\alpha )^\bullet $ is acyclic in degrees $\geq m$. Since $-\otimes _ A^{\mathbf{L}} B$ is an exact functor we get a distinguished triangle

\[ (E^\bullet \otimes _ A^{\mathbf{L}} B, K^\bullet \otimes _ A^{\mathbf{L}} B, C(\alpha )^\bullet \otimes _ A^{\mathbf{L}} B) \]

of complexes of $B$-modules. By the dual to Derived Categories, Lemma 13.17.1 we see that $H^ i(C(\alpha )^\bullet \otimes _ A^{\mathbf{L}} B) = 0$ for $i \geq m$. Since $E^\bullet $ is a complex of projective $R$-modules we see that $E^\bullet \otimes _ A^{\mathbf{L}} B = E^\bullet \otimes _ A B$ and hence

\[ E^\bullet \otimes _ A B \longrightarrow K^\bullet \otimes _ A^{\mathbf{L}} B \]

is a morphism of complexes of $B$-modules that witnesses the fact that $K^\bullet \otimes _ A^{\mathbf{L}} B$ is $m$-pseudo-coherent. The case of pseudo-coherent complexes follows from the case of $m$-pseudo-coherent complexes via Lemma 15.62.5. $\square$


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